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Let $i:M\hookrightarrow N$ be the inclusion. The restriction of a connection to an embedded submanifold is the pullback connection on $i^{\ast }TN=TN{|}_M$. Concretely, for a tangent vector field $X$ on $M$ and a local section $S$ of $TN{|}_M$, extend $S$ smoothly off $M$ and define ${\nabla }_{X}S$ by differentiating the extension in direction $di(X)$. Two extensions differ by a field whose coefficients vanish on $M$. Their derivatives along every tangent curve in $M$ vanish too, and the connection's coefficient terms are multiplied by the zero field. The answer is therefore independent of the extension. Dependence only on the value of the first argument follows from the connection's linearity over smooth functions.

The induced Riemannian metric is positive definite, so the fibrewise orthogonal projection $\pi :TN{|}_M\to TM$ is smooth. For tangent fields define the projected ambient connection $D_{X}Y=\pi ({\nabla }_{X}Y)$. It is real-linear and satisfiesbecause $\pi Y=Y$. Thus $D$ is a Koszul connection on $M$.

The Levi-Civita connection is characterized by the torsion-free connection and metric connection conditionsThese determine it uniquely. For the ambient Levi-Civita connection, the Gauss formula is ${\nabla }_{X}Y=D_{X}Y+II(X,Y)$, withLinearity over smooth functions in $X$ is immediate. In $Y$, the additional $X(f)Y$ term is tangent and is killed by $I-\pi$. Thus tensoriality makes $II$ a bilinear map $TM{\times }_{M}TM\to \nu M$ into the normal bundle. Its antisymmetric part isbecause the Lie bracket of vector fields tangent to $M$ is tangent. This proves the symmetry of the second fundamental form in an arbitrary Riemannian ambient manifold.

Write the curvature operators as $R^N(v,w)y={\nabla }_v{\nabla }_{w}y-{\nabla }_w{\nabla }_{v}y-{\nabla }_{[v,w]}y$ and $R^M$ with $D$ in place of $\nabla$. To match the four-slot notation in the requested identity, useThis convention is stated because permuting the slots can change the displayed signs.

If $\eta$ is a normal field and $x$ is tangent, metric compatibility applied to $⟨\eta ,x⟩=0$ gives the tangential derivative of a normal field identityUsing the Gauss formula and this identity,Here the normal part of ${\nabla }_v(D_{w}y)$ pairs to zero with $x$. Interchange $v,w$ and subtract; the bracket term obeys $⟨{\nabla }_{[v,w]}y,x⟩=⟨D_{[v,w]}y,x⟩$. It follows thatRearranging proves the Gauss equation in a curved ambient manifold:All expressions are tensorial, so choosing local extensions of the four tangent vectors proves the pointwise identity at $P$. With flat Euclidean ambient space, $R^N=0$ and this reduces to the usual Gauss equation.

Use $t$ as an affine parameter on each member of the geodesic variation, so ${\nabla }_{T}T=0$. The parameter directions commute, giving $[T,S]=0$ along the variation. The torsion-free connection then givesConsequently the covariant derivative and curvature definition giveThis is the geodesic-deviation equation with the stated curvature sign:The component expression defines the slot order: $\beta$ is the vector acted on, and $\gamma ,\delta$ are the two derivative directions of the Riemann curvature tensor. Both the commuting parameter directions and affine parametrization are essential; a general connecting vector field or a nonaffine parameter would add terms.

An affine connection on $M$ is an operation $(Y,X)\mapsto {\nabla }_{Y}X$ on smooth vector fields that is real-bilinear, linear over $C^{\infty }(M)$ in the differentiating field, and satisfiesIt is a connection on a vector bundle for the bundle $TM$, also called a linear connection on the manifold. Its curvature form of a connection has the operator formThis convention fixes its sign. Expanding the two connection product rules shows that the extra derivatives of a scalar cancel, so $R$ is linear over smooth functions in all three fields. For a Levi-Civita connection, this is the Riemann curvature tensor.

The torsion form is $T(Y,Z)={\nabla }_{Y}Z-{\nabla }_{Z}Y-[Y,Z]$. For a Riemannian metric $g$, its Levi-Civita connection is the unique torsion-free connection satisfying metric compatibilityExistence and uniqueness follow from the Koszul formula. In coordinates its coefficients areSymmetry in $b,c$ gives zero torsion, and substitution verifies metric compatibility.

Use $x^0=t$ and the Levi-Civita symbol ${ϵ}^1{}_{23}=1$. Define a smooth affine connection byThe lower indices are symmetric, so it is a torsion-free connection. With $t$ itself as parameter, the time component of the geodesic equation is identically satisfied, while its spatial components becomeThe last term is $2(\mathbf{B}\times \dot{\mathbf{x}}{)}^i$, with the wedge in the source interpreted as the three-dimensional cross product. Thus the spacetime lifts $(t,\mathbf{x}(t))$ are actually affinely parametrized geodesics, which proves the requested unparametrized claim. Conversely a geodesic with nonzero constant $dt/ds$ can be parametrized by $t$ and obeys this particle equation; purely spatial geodesics with $dt/ds=0$ are not such trajectories. No field equations or metric compatibility are needed. This construction is the affine connection for a velocity-linear force.

The Levi-Civita connection is a torsion-free connection. Its torsion form isSince $T=0$,In coordinates this is the symmetry ${\Gamma }_{jk}^i={\Gamma }_{kj}^i$ of the Christoffel symbols.

The solder form on $X$ is the $TX$-valued one-formIn a coordinate frame it is $\theta =d{x}^i\otimes {\partial }_i$. The torsion form of $\nabla$ isEquivalently, for vector fields $U,V$,Writingand using $[{\partial }_i,{\partial }_j]=0$ givesHence $\nabla$ is torsion-free exactly whenfor every $i,j,k$.

The induced dual connection is uniquely defined bywhich immediately gives the required pairing identity. Ifthen

A connection on $TB$ is symmetric, or torsion-free, whenequivalently ${\Gamma }_{ij}^k={\Gamma }_{ji}^k$ in coordinates. With $\mathrm{Alt}B(X,Y)=B(X,Y)-B(Y,X)$,

For a Riemannian metric on an ambient manifold and an embedded submanifold, let $\pi :TN{|}_M\to TM$ be orthogonal projection. Project the restriction of a connection to an embedded submanifold to obtain $D_{X}Y=\pi {\nabla }_{X}Y$. Since $\pi Y=Y$, the Leibniz rule and linearity over smooth functions hold, making $D$ an affine connection. If the ambient connection is the Levi-Civita connection, pairing with tangent vectors shows that $D$ is a metric connection; projecting the zero-torsion identity shows it is a torsion-free connection. By uniqueness, $D$ is the induced Levi-Civita connection.

Two torsion-free connections are projectively equivalent if they have the same unparametrized geodesics. Their difference of affine connections is a tensor $S$. A change of parameter adds only tangential acceleration, so equality of curves requires $S(v,v)$ to be parallel to $v$ for every tangent vector. The symmetric bilinear diagonal-parallel lemma gives exactly the displayed condition. Conversely the geodesic reparametrization under projective equivalence absorbs this tangential term. Agreement with the same affine parameters instead forces $S=0$.

The spinor covariant derivative has curvature $d\Omega +\Omega \wedge \Omega =\frac{1}{4}{R}_{AB}{\gamma }^{AB}$, since the spin representation preserves the Lorentz algebra bracket. In a coordinate basis with the torsion-free connection, antisymmetric second covariant derivatives therefore give the displayed commutator. With unit-weight antisymmetrization, ${\nabla }_{[\mu }{\nabla }_{\nu ]}\psi =\frac{1}{8}{R}_{\mu \nu AB}{\gamma }^{AB}\psi =\frac{1}{16}{R}_{\mu \nu AB}[{\gamma }^A,{\gamma }^B]\psi$. A noncoordinate frame requires its frame-commutator correction when using the operator commutator; the full second covariant derivative includes that correction automatically.

The second fundamental form of an embedded submanifold of a Riemannian manifold is symmetric. Its ambient Levi-Civita connection is a torsion-free connection, soThe Lie bracket of vector fields tangent to the submanifold is tangent and therefore has zero normal component. This proof applies to a curved ambient manifold as well as Euclidean space.

Antisymmetrization over selected tensor indices averages their permutations with the sign of a permutation. Square brackets denote it, for example $T_{[ab]}=(T_{ab}-T_{ba})/2$. It projects onto the alternating part. Antisymmetrized second derivatives of a scalar vanish for a torsion-free connection, but antisymmetrized later derivatives act on a tensor and can produce curvature.

The difference of two torsion-free connections is symmetric, because subtracting their zero-torsion identities gives $\Delta (X,Y)=\Delta (Y,X)$. Agreement of parametrized geodesics makes this difference skew-symmetric as well, by parametrized geodesics determine the symmetric part of an affine connection. Over the real numbers it must therefore vanish. Without the torsion-free condition, an arbitrary skew-symmetric difference changes torsion without changing the parametrized geodesics.